程序代写代做代考 game Example 1: Finding Mixed Strategy Nash Equilibrium

Example 1: Finding Mixed Strategy Nash Equilibrium
Find all the pure and mixed strategy equilibria of the following game by constructing the best response correspondences of the players:
Answer: First let us consider best responses to pure strategies
So the game has NO pure strategy Nash Equilibrium. Mixed Strategies: Consider a mixed strategy in which
• P1 chooses T with probability p and B with probability 1 p • P2 chooses L with probability q and R with probability 1 q Given player 2’s mixed strategy (q, 1 q),
if P1 plays T: expected payo↵ 2q + (1 q)0 = 2q ifP1playsB: expectedpayo↵ q+(1q)3=32q
For P1 to be indi↵erent between T and B in equilibrium, we must have
2q = 3 2q ) q = 3 4
L
R
T
2,1
0,2
B
1,2
3,0
BR1(L) = T
BR2(T) = R
BR1(R) = B
BR2(B) = L
8< T B i f q > 3 9= 4
if q < 3 :4; BR1 = Similarly, given P1’s mixed strategy (p, 1 p), we have for player 2: ifP2playsL: expectedpayo↵ p+(1p)2=2p ifP2playsR expectedpayo↵ 2p+(1p)0=2p indi↵erent between T and B if q = 3 4 For P2 to be indi↵erent between L and R, we must have 2 p = 2p ) p = 2 3 i f p < 2 9= 3 BR2((p, 1 p)) = Hence in the (unique) mixed strategy NE 8< L R if p > 2 :3;
indi↵erent between R and L if p = 2 3
• Player 1 plays T with probability 2 and B with probability 1 . 33
• Player 2 plays L with probability 3 and R with probability 1 . 44
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Example 2: Finding Mixed Strategy Nash Equilibrium
Find all the pure and mixed strategy equilibria of the following game by constructing the best response correspondences of the players:
Answer: First let us consider best responses to pure strategies
So the game has TWO pure strategy Nash Equilibria (Opera,Opera) and (Fight, Fight).
Mixed Strategies: Consider a mixed strategy in which
• P1 chooses O with probability p and F with probability 1 p • P2 chooses O with probability q and F with probability 1 q Given P2’s mixed strategy (q, 1 q), we have for player 1:
if P1 plays 0: expected payo↵ 2q + (1 q)0 = 2q ifP1playsF: expectedpayo↵ q0+(1q)1=1q
For player 1 to be indi↵erent between Opera and Fight in equilibrium, we must
have
BR1((q, 1 q)) =
Similarly, given player 1’s mixed strategy (p, 1 p),
ifP2plays0: expectedpayo↵ p+(1p)0=p ifP1playsF: expectedpayo↵ p0+(1p)2=22p
For P2 to be indi↵erent between Opera and Fight in equilibrium, we must have
Opera(O)
Fight (F)
Opera(O)
2,1
0,0
Fight (F)
0,0
1,2
BR1(O) = O
BR2(O) = O
BR1(F) = F
BR2(F) = F
2q = 1 q ) q = 1 839
1 = 3
Fight if q < 1 :3; indi↵erent between Opera and Fight if q = 1 3 p = 2 2p ) p = 2 8< O p e r a Fight 3 i f p > 2 9= 3
if p > 2 :3;
BR2((p, 1 p)) =
Hence in the (unique) mixed strategy NE
indi↵erent between Opera and Fight if p = 2 3
• Player 1 plays Opera with probability 2 and Fight with probability 1 . 33
• Player 2 plays Opera with probability 1 and Fight with probability 2 . 33
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